Subgroups — #26220498

Order 12032.

974 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245853	C₂	1	2	yes	maximal, complemented
2	#20244255	C₂	1	2	yes	maximal
3	#20244122	C₂	1	2	yes	maximal, complemented
4	#20244073	C₂	1	4	yes	maximal, complemented
5	C₄×#13093958	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13093945	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	—	2	2	no	—
10	C₂×#13093958	C₂²	1	2	yes	—
11	C₂×#13093945	C₂²	1	4	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
15	#15689026	C₂²	1	4	yes	—
16	#15689022	C₂²	1	4	yes	—
17	#15689016	—	2	4	no	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
20	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688886	C₂²	1	2	yes	—
22	#15688885	C₂²	1	4	yes	—
23	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
25	C₄×D₈×C₄₇	C₄	1	8	yes	complemented
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	#13093983	—	4	8	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
34	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	#13093958	—	2	4	no	—
41	#13093945	—	2	8	no	—
42	C₄×C₄₇⋊C₈	—	2	2	no	—
43	C₈×Dic₄₇	—	2	4	no	—
44	C₈×Dic₄₇	—	2	4	no	—
45	C₄₇⋊#95	—	2	2	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₄₇	—	2	2	no	—
49	C₂×D₈×C₄₇	—	2	4	no	—
50	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₄₇	—	2	4	no	—
54	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#107×C₄₇	C₂×C₄	1	4	yes	—
59	#102×C₄₇	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
61	C₂⁴×C₄₇	—	2	2	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	4	4	no	—
69	C₂²×C₄×C₄₇	—	4	4	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
75	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
76	D₈×C₄₇	—	2	8	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₇	#41	1	4	yes	—
81	C₄⋊C₄×C₄₇	—	2	2	no	—
82	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₄₇	—	2	8	no	—
84	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₇	#41	1	2	yes	—
86	C₂²⋊Dic₄₇	—	4	8	no	—
87	C₂²⋊Dic₄₇	—	4	8	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	4	8	no	—
91	C₄×Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	—	2	2	no	—
93	C₄×Dic₄₇	—	2	2	no	—
94	C₄×Dic₄₇	—	2	4	no	—
95	C₄×Dic₄₇	—	4	4	no	—
96	C₄×Dic₄₇	D₈	1	4	yes	complemented
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	#41	1	4	yes	—
100	C₂³×C₄₇	—	2	4	no	—
101	C₂³×C₄₇	—	4	4	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	—	4	4	no	—
104	C₂³×C₄₇	—	4	2	no	—
105	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₄₇	—	2	4	no	—
107	D₄×C₄₇	—	2	4	no	—
108	D₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	16	no	—
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₄₇	#135	1	2	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
130	C₂×Dic₄₇	#135	1	4	yes	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	#117	1	4	yes	complemented
133	C₄₇⋊C₈	—	2	4	no	—
134	#21275	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	4	4	no	—
136	C₂²×C₄₇	—	4	4	no	—
137	C₂²×C₄₇	—	4	4	no	—
138	C₂²×C₄₇	—	8	4	no	—
139	C₂²×C₄₇	—	8	4	no	—
140	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₄₇	#532	1	2	yes	—
142	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#135	1	2	yes	—
144	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	4	4	no	—
147	C₄×C₄₇	C₄×D₈	1	4	yes	—
148	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
149	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
150	C₄×C₄₇	#529	1	2	yes	—
151	Dic₄₇	—	2	4	no	—
152	Dic₄₇	—	2	4	no	—
153	Dic₄₇	—	8	4	no	—
154	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
155	#2293	—	47	2	no	—
156	#1970	—	47	2	no	—
157	#1942	—	47	2	no	—
158	#1934	—	47	4	no	—
159	C₄×#107	—	47	1	no	—
160	C₄×#102	—	47	2	no	—
161	C₄²×C₈	—	47	1	no	—
162	C₂×C₄₇	—	8	4	no	—
163	C₂×C₄₇	#2293	1	1	yes	characteristic
164	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
165	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
166	C₂×C₄₇	#3168	1	2	yes	—
167	C₂×C₄₇	#3169	1	2	yes	—
168	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
170	C₂×#117	—	94	2	no	—
171	C₄×D₈	Dic₄₇	1	8	yes	complemented
172	C₂×#107	—	47	2	no	—
173	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
174	C₂×#102	—	47	4	no	—
175	C₂×#102	C₂×D₄₇	1	2	yes	—
176	C₂×C₄×C₈	—	47	2	no	—
177	C₂×C₄×C₈	—	47	2	no	—
178	C₂×C₄×C₈	—	47	1	no	—
179	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
180	#390	—	47	4	no	—
181	#389	—	47	4	no	—
182	#388	—	94	4	no	—
183	C₄×C₄⋊C₄	—	47	2	no	—
184	C₄×C₂²⋊C₄	—	94	2	no	—
185	C₄³	—	47	1	no	—
186	#340	—	47	2	no	—
187	#340	—	47	4	no	—
188	C₄₇	#21275	1	1	yes	characteristic, complemented, Sylow
189	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
190	C₂³×C₄	—	2	2	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
193	C₂²×C₈	—	47	2	no	—
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	C₂×Dic₄₇	1	8	yes	—
198	#117	—	188	8	no	—
199	C₂×C₄⋊C₄	—	47	4	no	—
200	C₂×C₄⋊C₄	—	94	4	no	—
201	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	94	4	no	—
204	C₂×C₂²⋊C₄	—	94	4	no	—
205	C₂×C₄²	—	47	1	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	47	1	no	—
208	C₂×C₄²	—	47	2	no	—
209	C₂×C₄²	—	94	2	no	—
210	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
211	#107	—	94	4	no	—
212	#107	C₂×Dic₄₇	1	4	yes	—
213	#102	—	94	8	no	—
214	#102	C₂×Dic₄₇	1	8	yes	—
215	C₄×C₈	—	94	4	no	—
216	C₄×C₈	—	94	4	no	—
217	C₄×C₈	—	94	2	no	—
218	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
219	#95	—	94	4	no	—
220	#95	—	94	2	no	—
221	C₂⁴	—	2	2	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	47	2	no	—
231	C₂²×C₄	—	47	2	no	—
232	C₂²×C₄	—	94	4	no	—
233	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
236	D₈	—	2	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	47	4	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	—	94	2	no	—
241	C₂×C₈	#10625350	1	4	yes	—
242	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
243	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	94	4	no	—
246	C₄⋊C₄	—	94	4	no	—
247	C₄⋊C₄	—	188	8	no	—
248	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	188	8	no	—
251	C₂²⋊C₄	—	188	8	no	—
252	C₄²	—	47	4	no	—
253	C₄²	—	94	4	no	—
254	C₄²	—	94	2	no	—
255	C₄²	—	94	2	no	—
256	C₄²	—	94	2	no	—
257	C₄²	—	188	4	no	—
258	C₄²	#10625352	1	2	yes	—
259	C₄²	C₂²×Dic₄₇	1	2	yes	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	4	no	—
265	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	16	no	—
274	C₂×C₄	—	47	4	no	—
275	C₂×C₄	—	47	4	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	94	2	no	—
279	C₂×C₄	—	94	2	no	—
280	C₂×C₄	—	188	4	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	—	188	4	no	—
283	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
284	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
285	C₂×C₄	D₈×D₄₇	1	2	yes	—
286	C₂×C₄	#13094043	1	2	yes	—
287	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
288	C₂×C₄	#41×D₄₇	1	2	yes	—
289	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625350	1	2	yes	—
291	C₈	—	2	2	no	—
292	C₈	—	94	4	no	—
293	C₈	#13093983	1	4	yes	complemented
294	C₂²	—	4	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
300	C₂²	#15689670	1	2	yes	—
301	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
302	C₂²	#135×D₄₇	1	2	yes	—
303	C₂²	C₂×#13094043	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	4	no	—
306	C₄	—	94	4	no	—
307	C₄	—	94	4	no	—
308	C₄	—	376	4	no	—
309	C₄	D₈×Dic₄₇	1	4	yes	—
310	C₄	C₂×#13093983	1	1	yes	characteristic
311	C₄	#41×Dic₄₇	1	1	yes	characteristic
312	C₄	#15689655	1	2	yes	—
313	C₂	—	8	4	no	—
314	C₂	#20245853	1	1	yes	characteristic
315	C₂	C₂×D₈×Dic₄₇	1	1	yes	characteristic
316	C₂	#135×Dic₄₇	1	1	yes	characteristic
317	C₂	#20253956	1	2	yes	—
318	C₂	#20253964	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#21275
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20245853	C₂	semidirect
3. #20244122	C₂	semidirect
4. #20244073	C₂	semidirect
5. C₄×#13093958	C₂	semidirect
6. C₄×#13093945	C₂	semidirect
7. C₄×C₈×Dic₄₇	C₂	semidirect
25. C₄×D₈×C₄₇	C₄	semidirect
96. C₄×Dic₄₇	D₈	semidirect
132. C₈×C₄₇	#117	semidirect
171. C₄×D₈	Dic₄₇	semidirect
188. C₄₇	#21275	semidirect
293. C₈	#13093983	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20245853	2
#20244255	2
#20244122	2
#20244073	2
C₄×#13093958	2
C₄×#13093945	2
C₄×C₈×Dic₄₇	2
C₂×C₄×D₈×C₄₇	2
#21275	47